..
zn = a0 + a1g1(xn) + ?· ?· ?· + amgm(xn)
?† An = (1 gi(xj))+
i?‰¤m,j?‰¤n??«??¬??
X1
...
Xn
??¶??·???
= M+
n ??«??¬??
Z1
...
Zn
??¶??·???
. (4)
In equation (4) gi(x) are nonlinear basis functions that depend on the
sample location x, while the field variables zi represent the field model at
location xi. Because the pseudo-inverse of the matrix Mn = (1 gi(xj))i?‰¤m,j?‰¤n
can be directly calculated from M+
n = (MT
n Mn)??’1MT
n , we obtain a closedform
solution for the covariance of the unknown parameters aj :
?† An =
??«??¬??¬??¬??¬??¬??¬??
n
Xj=1
(1 ?· ?· ?· gi(xj) ?· ?· ?· gm(xj))
??«??¬??¬??¬??¬??¬??¬??
1
...
gi(xj)
...
gm(xj)
??¶??·??·??·??·??·??·???
??¶??·??·??·??·??·??·???
??’1
n
Xj=1
Zj
??«??¬??¬??¬??¬??¬??¬??
1
...
gi(xj)
...
gm(xj)
??¶??·??·??·??·??·??·???
. (5)
The covariance matrix of ?† An can now be related directly to the (constant)
measurement uncertainty as: var( ?† An) = var(zi)(MT
n Mn)??’1 , and the adaptive
sampling algorithm will move the vehicle from location xn to xn+1, such that
the following p-norm is maximized over the search space ??–:
m(x) =
?°?°?°?°?°?°?°?°?°?°?°
??«??¬??¬??¬??¬??¬??
1
...
MT
n gi(x)
...
gm(x)
??¶??·??·??·??·??·???
?µ 1
1 ?· ?· ?· gi(x) ?· ?· ?· gm(x)?¶
?°?°?°?°?°?°?°?°?°?°?°
,
43
Dan O. Popa and Frank L. Lewis
m(xn+1) ?‰? m(x), (???)x ??? ??–. (6)
A summary of the deployment algorithm using closed-form variance estimation
is shown below:
Closed-Form Adaptive Sampling Algorithm (AS-1)
Assumptions: The field model is linear in the parameters, field model
functions gj are known, sampling locations xj belong to a discrete sampling
space ??– and do not have uncertainty.
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